The matrix \(M:=M_1\)  in Lemma REF  encodes several important numerical invariants of \(B:=B_1\) . Obviously, the size of \(M\)  is \(k(B)\times k(B)\)  where \(k(B):=|\operatorname{Irr}(B)|\) . By (REF ), the rank of \(M\)  equals \(l(B):=|\operatorname{IBr}(B)|\) . Recall that the height \(h(\chi )\ge 0\)  of \(\chi \in \operatorname{Irr}(B)\)  is defined by \(\chi (1)_p=p^{a-d+h(\chi )}\)  where \(|G|_p=p^a\)  and \(d\)  is the defect of \(B\) . If \(h(\chi )=0\) , then the \(p\) -adic valuation of \([\chi ,\psi ]^0\)  equals \(h(\psi )-d\)  for every \(\psi \in \operatorname{Irr}(B)\)  (see [1]}). Moreover, if \(h(\chi )>0\) , then the \(p\) -adic valuation of \([\chi ,\chi ]^0\)  is larger than \(-d\)  (see [1]}).
In this way, \(M\)  encodes all character heights. Finally, one can show that \(M\)  determines the elementary divisors of the Cartan matrix of \(B\) . However, the next theorem implies that all these invariants do not suffice to distinguish blocks of symmetric groups.
